Wind Energy Handbook. Michael Barton Graham

reversal and breakdown into turbulence in the wake of a porous plate occurs, typically starting when the resistance coefficient K (= Δp/(½ρU²)) exceeds 4, experimental measurements show that the axial force on the body departs from the well‐known theory of Taylor (1944) for ordered flow through a porous plate. Similarly, experimental measurements of the thrust force coefficient for a rotor – for example, reported by Glauert (1926) and plotted in Figure 3.16 – show a departure from the actuator disc momentum theory C_T = 4a(1 − a). In both cases the measured forces are larger than the predictions of theory, and in both cases the point of break‐away is near the maximum predicted by the momentum theory.

Figure 3.16 Comparison of theoretical and measured values of C_T.

      The thrust (or drag) coefficient for a simple, flat circular plate is given by Hoerner (1965) as 1.17 but, as demonstrated in Figure 3.16, the thrust on the rotor reaches a higher value. A major difference between the wake of the circular plate and of the rotor is that the latter contains a strong rotating component even after flow reversal in the wake has started.

      It would follow from the above arguments that for high values of the axial induction factor a large part of the pressure drop across the disc is not simply associated with blade circulation, just as it is absent in the case of the circular plate. Circulation would cause a pressure drop similar to that given by the momentum theory determined by the very low axial velocity of the flow that actually permeates the disc.

      3.7.3 Empirical determination of thrust coefficient

      A suitable straight line through the experimental points would appear to be possible, although Glauert proposed a parabolic curve, and provides an empirical solution to the problem of the thrust on a heavily loaded turbine (a rotor operating at a high value of the axial flow induction factor).

      Most authors assume that the entire thrust on the rotor disc is associated with axial momentum change. Therefore, for the empirical line to be useful it must be assumed that it applies not only to the whole rotor but also to each separate streamtube. Let C_T1 be the empirical value of C_T when a = 1. Then, as the straight line must be a tangent to the momentum theory parabola at the transition point, the equation for the line is

(3.58)

      and the value of a at the transition point is

By inspection, C_T1 must lie between 1.6 and 2: C_T1 = 1.816 would appear to be the best fit to the experimental data of Figure 3.16, whereas Wilson et al. (1974) favour the lower value of C_T1 = 1.6. Glauert fits a parabolic curve to the data [replacing a in the mass flow expression by 4a(1 − a)/(0.6 + 0.61a + 0.79a²) when a > 1/3] giving much higher values of C_T1 at high values of a but he was considering the case of an airscrew in the windmill brake state where the angles of attack are negative. De Vaal et al. (2014) suggest a be replaced by 0.25a(5 − 3a), similarly giving a somewhat lower windmill brake state result.

      The flow field through the turbine under heavily loaded conditions cannot be modelled easily, and the results of this empirical analysis must be regarded as being only approximate at best. They are, nevertheless, better than those predicted by the momentum theory. For most practical designs the value of the axial flow induction factor rarely exceeds 0.6 and for a well‐designed rotor will be in the vicinity of 0.33 for much of its operational range.

For values of a greater than a_T, it is common to replace the momentum theory thrust in Eq. (3.9) with Eq. (3.58), in which case Eq. (3.54a) is replaced by

(3.59)

      However, as the additional pressure drop is caused by breakdown of the streamline wake, this course of action is questionable, and it may be more appropriate to retain Eq. (3.54).

3.8 Blade geometry

      3.8.1 Introduction

      The purpose of most wind turbines is to extract as much energy from the wind as possible, and each component of the turbine has to be optimised for that goal. Optimal blade design is influenced by the mode of operation of the turbine, that is, fixed rotational speed or variable rotational speed and, ideally, the wind distribution at the intended site. In practice engineering compromises are made, but it is still necessary to know what would be the best design.

Optimising a blade design means maximising the power output, and so a suitable solution to BEM Eqs. (3.54 or (3.59) and (3.55)) is necessary.

      3.8.2 Optimal design for variable‐speed operation

      A turbine operating at variable speed can maintain the constant tip speed ratio required for the maximum power coefficient to be developed regardless of wind speed. To develop the maximum possible power coefficient requires a suitable blade geometry, the conditions for which will now be derived.

      For a chosen tip speed ratio λ the torque developed at each blade station is given by Eq. (3.49) and is maximised if

      giving

(3.60)

From Eqs. (3.51) and Скачать книгу